I have the following characteristic function of S.
I understood how I got to this characteristic function yet the next part I want to show that as $$ n \rightarrow \infty, \varphi_S (t) \rightarrow e^{-\frac{t^2}{2}} $$
The solution states:
I am very confused how to get to even the first line. I feel like it's something to do with Taylor series yet I am unsure.
When $n$ is large, the arguments of the exponentials are small, allowing a limited Taylor development, and
$$pe^{ka}+(1-p)e^{-{a/k}}\approx p\left(1+ka+\frac{k^2a^2}2\right)+(1-p)\left(1-\frac ak+\frac{a^2}{2k^2}\right).$$
The constant term is $1$. The linear coefficient is
$$\frac{pk^2-(1-p)}ka=0.$$
Then the quadratic coefficient (which I let you evaluate exactly) will be proportional to $a^2$, and of the form $\dfrac{bt^2}n$. Finally, by a well-known formula,
$$\lim_{n\to\infty}\left(1+\frac{bt^2}{n}\right)^n=e^{bt^2}.$$
Higher order terms in the development will decrease faster and bring no contribution to the limit.